Functional equations and harmonic extensions
Lowell J. Hansen, Harold S. Shapiro · Complex Variables Theory and Application An International Journal · 1994
In this paper, we first consider the posibility of extending to the exterior of a region the solution to a Dirichlet problem. We limit ourselves to the situation where the boundary curve is analytic and the boundary data is polynomial or real-entire. We then turn to the problem of hte existence of solutions to the functional equation f(P(z))-g(Q(z)) = k(z),, where P and Q are given polynomials and k is a given meromorphic function. Central to our results is a generalization of a theorem of A. and C. Rényiconcerning periodic functions of the form f(P). We conclude by showing how the two problems ae related.